Differentials of One Variable

• assignment Zapping With d 1

  assignment Homework

  Zapping With d 1

  Energy and Entropy 2021 (2 years)

  Find the differential of each of the following expressions; zap each of the following with \(d\):

  1. \[f=3x-5z^2+2xy\]

  2. \[g=\frac{c^{1/2}b}{a^2}\]

  3. \[h=\sin^2(\omega t)\]

  4. \[j=a^x\]

  5. \[k=5 \tan\left(\ln{\left(\frac{V_1}{V_2}\right)}\right)\]

• group Systems of Equations Compare and Contrast

  group Small Group Activity

  60 min.

  Systems of Equations Compare and Contrast

• assignment Approximating a Delta Function with Isoceles Triangles

  assignment Homework

  Approximating a Delta Function with Isoceles Triangles

  Static Fields 2023 (6 years)

  Remember that the delta function is defined so that \[ \delta(x-a)= \begin{cases} 0, &x\ne a\\ \infty, & x=a \end{cases} \]

  Also: \[\int_{-\infty}^{\infty} \delta(x-a)\, dx =1\].

  1. Find a set of functions that approximate the delta function \(\delta(x-a)\) with a sequence of isosceles triangles \(\delta_{\epsilon}(x-a)\), centered at \(a\), that get narrower and taller as the parameter \(\epsilon\) approaches zero.
  2. Using the test function \(f(x)=3x^2\), find the value of \[\int_{-\infty}^{\infty} f(x)\delta_{\epsilon}(x-a)\, dx\] Then, show that the integral approaches \(f(a)\) in the limit that \(\epsilon \rightarrow 0\).

• assignment Flux through a Plane

  assignment Homework

  Flux through a Plane

  Static Fields 2023 (4 years) Find the upward pointing flux of the vector field \(\boldsymbol{\vec{H}}=2z\,\boldsymbol{\hat{x}} +\frac{1}{x^2+1}\boldsymbol{\hat{y}}+(3+2z)\boldsymbol{\hat{z}}\) through the rectangle \(R\) with one edge along the \(y\) axis and the other in the \(xz\)-plane along the line \(z=x\), with \(0\le y\le2\) and \(0\le x\le3\).
• group Fourier Transform of a Shifted Function

  group Small Group Activity

  5 min.

  Fourier Transform of a Shifted Function

  Periodic Systems 2022

  Fourier Transforms and Wave Packets

• assignment The puddle

  assignment Homework

  The puddle

  differentials Static Fields 2023 (5 years) The depth of a puddle in millimeters is given by \[h=\frac{1}{10} \bigl(1+\sin(\pi xy)\bigr)\] Your path through the puddle is given by \[x=3t \qquad y=4t\] and your current position is \(x=3\), \(y=4\), with \(x\) and \(y\) also in millimeters, and \(t\) in seconds.

  1. At your current position, how fast is the depth of water through which you are walking changing per unit time?
  2. At your current position, how fast is the depth of water through which you are walking changing per unit distance?
  3. FOOD FOR THOUGHT (optional)
     There is a walkway over the puddle at \(x=10\). At your current position, how fast is the depth of water through which you are walking changing per unit distance towards the walkway.
• assignment Find Area/Volume from $d\vec{r}$

  assignment Homework

  Find Area/Volume from \(d\vec{r}\)

  Static Fields 2023 (5 years)

  Start with \(d\vec{r}\) in rectangular, cylindrical, and spherical coordinates. Use these expressions to write the scalar area elements \(dA\) (for different coordinate equals constant surfaces) and the volume element \(d\tau\). It might help you to think of the following surfaces: The various sides of a rectangular box, a finite cylinder with a top and a bottom, a half cylinder, and a hemisphere with both a curved and a flat side, and a cone.

  1. Rectangular: \begin{align} dA&=\\ d\tau&= \end{align}
  2. Cylindrical: \begin{align} dA&=\\ d\tau&= \end{align}
  3. Spherical: \begin{align} dA&=\\ d\tau&= \end{align}

• assignment Rubber Sheet

  assignment Homework

  Rubber Sheet

  Energy and Entropy 2021 (2 years)

  Consider a hanging rectangular rubber sheet. We will consider there to be two ways to get energy into or out of this sheet: you can either stretch it vertically or horizontally. The distance of vertical stretch we will call \(y\), and the distance of horizontal stretch we will call \(x\).

  If I pull the bottom down by a small distance \(\Delta y\), with no horizontal force, what is the resulting change in width \(\Delta x\)? Express your answer in terms of partial derivatives of the potential energy \(U(x,y)\).

• format_list_numbered Integration Sequence

  format_list_numbered Sequence

  Integration Sequence

  Students learn/review how to do integrals in a multivariable context, using the vector differential \(d\vec{r}=dx\, \hat{x}+dy\, \hat{y}+dz\, \hat{z}\) and its curvilinear coordinate analogues as a unifying strategy. This strategy is common among physicists, but is NOT typically taught in vector calculus courses and will be new to most students.
• assignment Cube Charge

  assignment Homework

  Cube Charge

  charge density

  Integration Sequence

  Static Fields 2023 (6 years)

  1. Charge is distributed throughout the volume of a dielectric cube with charge density \(\rho=\beta z^2\), where \(z\) is the height from the bottom of the cube, and where each side of the cube has length \(L\). What is the total charge inside the cube? Do this problem in two ways as both a single integral and as a triple integral.
  2. On a different cube: Charge is distributed on the surface of a cube with charge density \(\sigma=\alpha z\) where \(z\) is the height from the bottom of the cube, and where each side of the cube has length \(L\). What is the total charge on the cube? Don't forget about the top and bottom of the cube.
• Static Fields 2023 (6 years) Find the total differential of the following functions:
  1. \(y=3x^2 + 4\cos 2x\)
  2. \(y=3x^2\cos kx\) (where \(k\) is a constant)
  3. \(y=\frac{\cos 7x}{x^2}\)
  4. \(y=\cos(3x^2-2)\)